Book:Haïm Brezis/Functional Analysis, Sobolev Spaces and Partial Differential Equations
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Haïm Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations
Published $\text {2010}$, Springer
- ISBN 978-0387709130
Subject Matter
Contents
Preface
- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions
- 1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functionals
- 1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets
- 1.3 The Bidual $E^{**}$. Orthogonality Relations
- 1.4 A Quick Introduction to the Theory of Conjugate Convex Functions
- Comments on Chapter 1
- Exercises for Chapter 1
- 2. The Uniform Boundedness Principle and the Closed Graph Theorem
- 2.1 The Baire Category Theorem
- 2.2 The Uniform Boundedness Principle
- 2.3 The Open Mapping Theorem and the Closed Graph Theorem
- 2.4 Complimentary Subspaces. Right and Left Invertability of Linear Operators
- 2.5 Orthogonality revisited
- 2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint
- 2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators
- Comments on Chapter 2
- Exercises for Chapter 2
- 3. Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity
- 3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous
- 3.2 Definition and Elementary Properties of the Weak Topology $\map \sigma {E,E^*}$
- 3.3 Weak Topology, Convex Sets and Linear Operators
- 3.4 The Weak* Topology $\map \sigma {E^*,E}$
- 3.5 Reflexive Spaces
- 3.6 Separable Spaces
- 3.7 Uniformly Convex Spaces
- Comments on Chapter 3
- Exercises for Chapter 3
- 4. $L^p$ Spaces
- 4.1 Some Results about Integration That Everyone Must Know
- 4.2 Definition and Elemenary Properties of $L^p$ Spaces
- 4.3 Reflexivity. Separability. Dual of $L^p$
- 4.4 Convolution and regularization
- 4.5 Criterion for Strong Compactness in $L^p$
- Comments on Chapter 4
- Exercises for Chapter 4
- 5. Hilbert Spaces
- 5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set
- 5.2 The Dual Space of a Hilbert Space
- 5.3 The Theorems of Stampacchia and Lax-Milgram
- 5.4 Hilbert sums. Orthonormal Bases
- Comments on Chapter 5
- Exercises for Chapter 5
- 6. Compact Operators, Spectral Decomposition of Self-Adjoint Compact Operators
- 6.1 Definitions. Elementary Properties. Adjoint
- 6.2 The Riesz-Fredholm Theory
- 6.3 The Spectrum of a Compact Operator
- 6.4 Spectral Decomposition of Self-Adjoint Compact Operators
- Comments on Chapter 6
- Exercises for Chapter 6
- 7. The Hille-Yosida Problem
- 7.1 Definition and Elementary Properties of Maximal Monotone Operators
- 7.2 Solution of the Evolution Problem $\dfrac {\d u} {\d t} + A u = 0$ on $\hointr 0 {+ \infty}$, $\map u 0 = u_0$. Existence and uniqueness
- 7.3 Regularity
- 7.4 The Self-Adjoint Case
- Comments on Chapter 7
- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension
- 8.1 Motivation
- 8.2 The Sobolev Space $\map {W^{1,p} } {I}$
- 8.3 The Space $W_0^{1,p}$
- 8.4 Some Examples of Boundary Value Problems
- 8.5 The Maximum Principle
- 8.6 Eigenfunctions and Spectral Decomposition
- Comments on Chapter 8
- Exercises for Chapter 8
- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in $N$ Dimensions
- 9.1 Definition and Elementary Properties of the Sobolev Spaces $\map {W^{1,p}} {\Omega}$
- 9.2 Extension Operators
- 9.3 Sobolev Inequalities
- 9.4 The Space $\map {W_0^{1,p} } \Omega$
- 9.5 Variational Formulation of Some Boundary Value Problems
- 9.6 Regularity of Weak Solutions
- 9.7 The Maximum Principle
- 9.8 Eigenfunctions and Spectral Decomposition
- Comments on Chapter 9
- 10. Evolution Problems: the Heat Equation and the Wave Equation
- 10.1 The Heat Equation: Existence, Uniqueness, and Regularity
- 10.2 The Maximum Principle
- 10.3 The Wave Equation
- Comments on Chapter 10
- 11. Miscellaneous Complements
- 11.1 Finite-Dimensional and Finite-Codimensional Spaces
- 11.2 Quotien Spaces
- 11.3 Some Classical Spaces and Sequences
- 11.4 Banach Space over $\C$: What Is Similar and What Is Different
Solutions of Some Exercises
Problems
Partial Solutions
Notation
References
Index